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WHEN DOES A CENTRAL BANK’S BALANCE SHEET REQUIREFISCAL SUPPORT?

MARCO DEL NEGRO AND CHRISTOPHER A. SIMS

ABSTRACT. The large balance sheet of many central banks has raised con-

cerns that they could suffer significant losses if interest rates rose, and

might need a capital injection from the fiscal authority. This paper con-

structs a simple deterministic general equilibrium model, calibrated to US

data, to study the the impact of alternative interest rates scenarios on the

central bank’s balance sheet. We show that the central bank’s policy rule

(or, equivalently, inflation objectives), and the behavior of seigniorage un-

der high inflation are crucial in determining whether a capital injection

might be needed. We show also that a central bank that is seen as ready, in

order to avoid a need for capital injection, to create seignorage by altering

its inflation objectives, can thereby lose control of the price level.

We conclude that for current balance sheet levels of the Federal Reserve

system, direct treasury support would be necessary only under what seem

rather extreme scenarios. However this result depends on assumptions

about demand for non-interest bearing Fed liabilities (mainly currency)

that cannot be firmly grounded in empirical estimates. Higher balance

sheet levels, or a lower currency demand than assumed here could force

the central bank to request a capital injection in order to maintain its infla-

tion control policy.

Date: April 14, 2014; First Draft: April 2013.

PRELIMINARY AND INCOMPLETE DRAFT

Marco Del Negro, Federal Reserve Bank of New York, [email protected].

Christopher A. Sims, Princeton University, [email protected]

We thank Seth Carpenter, and seminar participants at the Bank of Canada, ECB, and FRB

Philadelphia for very helpful comments.The views expressed in this paper do not neces-

sarily reflect those of the Federal Reserve Bank of New York or the Federal Reserve System.1

JEL CLASSIFICATION: E58, E59

KEY WORDS: central bank’s balance sheet, solvency

I. INTRODUCTION

Hall and Reis (2013) and Carpenter, Ihrig, Klee, Quinn, and Boote (2013)

have explored the likely path of the Federal Reserve System’s balance sheet

during a possible return to historically normal levels of interest rates. Both

conclude that, though a period when the system’s net worth at market value

is negative might occur, this is unlikely, would be temporary and would not

create serious problems.1 Those conclusions rely on extrapolating into the

future not only a notion of historically normal interest rates, but also of

historically normal relationships between interest rates, inflation rates, and

components of the System’s balance sheet. In this paper we look at com-

plete, though simplified, economic models in order to study why a central

bank’s balance sheet matters at all and the consequences of a lack of fiscal

backing for the central bank. These issues are important because they lead

us to think about unlikely, but nonetheless possible, sequences of events

that could undermine economic stability. As recent events should have

taught us, historically abnormal events do occur in financial markets, and

understanding in advance how they can arise and how to avert or mitigate

them is worthwhile.2

Constructing a model that allows us to address these issues requires us

to specify monetary and fiscal policy behavior and to consider how demand

1Christensen, Lopez, and Rudebusch (2013) study the interest rate risk faced by the Fed-

eral Reserve using probabilities for alternative interest rate scenarios obtained from a dy-

namic term structure model. They reach the similar conclusions as Hall and Reis (2013)

and Carpenter, Ihrig, Klee, Quinn, and Boote (2013).2A number of recent papers, including Corsetti and Dedola (2012) and Bassetto and

Messer (2013), also study the central bank’s and the fiscal authorities’ balance sheets

separately.

CENTRAL BANK’S BALANCE SHEET. PRELIMINARY DRAFT 1

for non-interest-bearing liabilities of the central bank (like currency, or re-

quired reserves paying zero interest) responds to interest rates. Results are

sensitive to these aspects of the specification, even when we consider a ver-

sion of the model calibrated to the current situation in the US.

In the first section below we consider a stripped down model to show

how the need for fiscal backing arises. In subsequent sections we make the

model more realistic and calibrate it to allow simulation of the US Federal

Reserve System’s response to realistic shifts in the real rate or “inflation

scares”.

In both the simple model and the more realistic one, we make some of

the same generic points.

• No policy undertaken by a central bank alone, without fiscal powers,

can guarantee a uniquely determined price level. Cochrane (2011)

has made this point carefully.

• There are simple, plausible, “backstop” fiscal and monetary policy

actions that will make the price level determinate. A commitment

to such actions eliminates the need for overt fiscal backing on the

economy’s equilibrium path.3

• When the price level is uniquely determined, it is nonetheless possi-

ble that the central bank, if its balance sheet is sufficiently impaired,

may need recapitalization in order to maintain its commitment to a

policy rule or an inflation target.

• It is important to distinguish the fiscal “backing” needed to fix the

price level from the fiscal “support” that a separate central bank may

need in order to implement its policy.

3Cochrane has argued that such actions are either impossible, or necessarily so drastic

as to be implausible. One contribution of this paper is to provide a counterexample to

Cochrane’s claim.


• A central bank’s ability to earn seigniorage can make it possible for

it to recover from a situation of negative net worth at market value

without recapitalization from the treasury, while still maintaining its

policy rule. Whether it can do so depends on the policy rule, the de-

mand for its non-interest-bearing liabilities, and the size of the initial

net worth gap.

In the simple model we aim to explain qualitatively how the need for back-

ing or support can arise, while in the more realistic model we try to deter-

mine how likely it is that the US Federal Reserve System will need fiscal

support if interest rates return to more historically normal levels in the near

future.

II. THE SIMPLE MODEL

A central bank that issues fiat money requires fiscal backing if it is to

control the price level. To understand why, and how the need for fiscal

backing might manifest itself, we first consider a stripped-down model to

illustrate the principles at work.

A representative agent solves

maxC,B,M,F

∫ ∞

0e−βt log(Ct) dt subject to (1)

Ct · (1 + ψ(vt)) +B + M

Pt+ τt + Ft = ρFt +

rtBt

Pt+ Yt , (2)

where C is consumption, B is instantaneous nominal bonds paying interest

at the rate r, M is non-interest-bearing money, ρ is a real rate of return on

a real asset F, Y is endowment income, and τ is the primary surplus (or

simply lump-sum taxes, since we have no explicit government spending in

this model). Velocity vt is given by

vt =PtCt

Mt, vt ≥ 0, (3)

and the function ψ(.), ψ′(.) > 0 captures transaction costs.


The government budget constraint is

B + MPt

+ τt =rtBt

Pt. (4)

Monetary policy is an interest-smoothing Taylor rule:

r = θr ·(

r + θπ

( PP− π

)− r)

. (5)

The “Taylor Principle”, that θπ should exceed one, is the usual prescription

for “active” monetary policy.

First order conditions for the private agent are

∂C :1C

= λ(1 + ψ + ψ′v) (6)

∂F : − ˆλ = λ(ρ− β) (7)

∂B : −ˆλP+ β

λ

P+

λ

P

ˆPP= r

λ

P(8)

∂M : −ˆλP+ β

λ

P+

λ

P

ˆPP= ψ′v2 (9)

The ˆzt notation means the time derivative of the future expected path of z

at t. It exists even at dates when z has taken a jump, so long as its future

path is right-differentiable. Below we also use the d+dt operator for the same

concept.

We are taking the real rate ρ as exogenous, and in this simple version of

the model constant. The economy is therefore being modeled as either hav-

ing a constant-returns-to scale investment technology or as having access to

international borrowing and lending at a fixed rate. Though we could ex-

tend the model to consider stochastically evolving Y, ρ, and other external

disturbances, here we consider only surprise shifts at the t = 0 starting date,

with perfect-foresight deterministic paths thereafter. This makes it easier

to follow the logic, though it makes the time-0 adjustments unrealistically

abrupt.


Besides the exogenous influences that already appear explicitly in the

system above (ρ and Y), we consider an “inflation scare” variable x. This

enters the agents’ first order condition as a perturbation to inflation expec-

tations. It can be reconciled with rational expectations by supposing that

agents think there is a possibility of discontinuous jumps in the price level,

with these jumps arriving as a Poisson process with a fixed rate. This would

happen if at these jump dates monetary policy created discontinuous jumps

in M. Such jumps would create temporary declines in the real value of gov-

ernment debt B/P which might explain why such jumps are perceived as

possible. If the jump process doesn’t change after a jump occurs, there is no

change in velocity, the inflation rate, consumption, or the interest rate at the

jump dates. Rather than solve a model that includes such jumps, we model

one in which the public is wrong about this — there are no jumps, despite

the expectation that there could be jumps. After a long enough period with

no jumps, the public would probably change its expectations, but there is

no logical contradiction in supposing that for a moderate amount of time

the fact that there are no jumps does not change expectations. In fact, if we

consider time-varying paths for x, in which x returns to zero after some pe-

riod, there is no way to distinguish whether the “true” model is one with

the assumed x path (and thus a non-zero probability of jumps in P) or one

with x ≡ 0 if jumps do not actually occur.

The inflation scare variable changes the first order conditions above to

give us

∂B : − ˆλ1P+ β

λ

P+

λ

P

(ˆPP+ x

)= r

λ

P(8’)

∂M : − ˆλ1P+ β

λ

P+

λ

P

(ˆPP+ x

)= ψ′v2 (9’)


Because the price and money jumps have no effect on interest rates or con-

sumption, no other equations in the model need change. These first or-

der conditions reflect the private agents’ use of a probability model that

includes jumps in evaluating their objective function.

We can solve the model analytically to see the impact of an unantici-

pated, permanent, time-0 shift in ρ (the real rate of return), x (the inflation

scare variable), or ρ (the central bank’s interest-rate target). We also solve

an expanded version below numerically for given exogenous time paths of

those variables.

Solving to eliminate the Lagrange multipliers from the first order condi-

tions we obtain

ρ = r−ˆPP− x (10)

r = ψ′(v)v2 (11)

− d+

dt

(1

C · (1 + ψ + ψ′v)

)=

ρ− β

C(1 + ψ + ψ′v). (12)

Using (10) and the policy rule (5), we obtain that along the path after the

initial date,

r = θr · ((θπ(r− ρ− x)− r + r− θππ)

= θr · (θπ − 1)r− θrθπ(ρ + x) + θr(r− θππ) . (13)

With the usual assumption of active monetary policy, θπ > 1, so this is an

unstable differential equation in the single endogenous variable r. Solutions

are of the form

rt = Et

[∫ ∞

0e−(θπ−1)θrsθrθπ(ρt+s + xt+s) ds

]− r− θππ

θπ − 1+ κe(θπ−1)θrt. (14)

In a steady state with x and ρ constant (and κ = 0), this give us

r =θπ(ρ + x)

θπ − 1− r− θππ

θπ − 1. (15)


From (11) we can find v as a function of r. Substituting the government bud-

get constraint into the private budget constraint gives us the social resource

constraint

C · (1 + ψ(v)) + F = ρF + Y. (16)

Solving this unstable differential equation forward gives us

Ft = Et

[∫ ∞

0exp

(−∫ s

0ρt+v dv

)(Yt+s − Ct+s(1 + ψ(vt+s))) ds

]. (17)

Here we do not include an exponentially explosive term because that would

be ruled out by transversality in the agent’s problem and by a lower bound

on F. With constant ρ, x and Y, r and therefore v are constant, and (12)

therefore lets us conclude that C grows (or shrinks) steadily at the rate ρ− β.

We can therefore use (17) to conclude that along the solution path, since ρ,

Y and v are constant

Ct =β · (ρ−1Y + Ft)

1 + ψ(v). (18)

This lets us determine initial C0 from the F0 at that date. From then on Ct

grows or shrinks at the rate ρ − β and the resulting saving or dissaving

determines the path of Ft from (18).

III. UNSTABLE PATHS, UNIQUENESS, FISCAL BACKING

To this point we have not introduced a central bank balance sheet or

budget constraint. We can nonetheless distinguish between monetary pol-

icy, controlling the interest rate or the money stock, and fiscal policy, con-

trolling the level of primary surpluses. Passive fiscal policies make the pri-

mary surplus co-move positively with the level of real debt and guarantee

a stable level of real debt, regardless of the time path of prices, under the as-

sumption of stable real interest rates. Passive fiscal policies generally leave

the price level indeterminate, no matter what interest rate or money stock

policy is in place. Guaranteeing uniqueness of the price level requires a

commitment to active fiscal policy. Nonetheless fiscal policy in the presence


of low inflation may be passive, so long as it is believed that policy would

turn active if necessary to rule out explosive inflation. In the remainder of

this section we make these points analytically in the simple model.

Our solution for r, given by (14), tells us that r could be constant, but

nothing in the model to this point tells us that κ 6= 0 is impossible. To

assess whether these paths are potential equilibria in the model, we need

to specify fiscal policy. The standard sort of fiscal policy to accompany the

type of monetary policy we have postulated (Taylor rule with θπ > 1) is

a “passive” policy that makes primary surpluses plus seigniorage respond

positively to the level of real debt. For example, we can assume

MP

+ τ = −φ0 + φ1BP

. (19)

Substituting this into the government budget constraint (4) and using (10)

gives us

b =

(ρ + x +

ˆP− PP− φ1

)b + φ0 . (20)

On an equilibrium path,ˆPP=

PP

,

that is, actual inflation and model-based expected inflation are equal. Thus

if φ1 > ρ + x, this is a stable differential equation, with b converging to

φ0/(φ1 − ρ − x). In fact, any φ1 > 0 is consistent with equilibrium, even

though for small values b grows exponentially. The transversality condition

with respect to debt for the private agent who holds the debt is

E0

[e−βt λB

Pt

]= 0 . (21)

From (7) λ grows at the rate β− ρ, while from (20) b grows asymptotically

at ρ + x− φ1. However the E0 in the transversality condition is the private

agent’s expectation operator. Since the agent believes in the possibility of

price jumps, the agent thinks that the expected real return on real debt is

just ρ, not ρ + x. Thus the agent believes that b grows asymptotically at the


rate ρ − φ1. The agent’s transversality condition is therefore satisfied for

any φ1 > 0. The agent in such equilibria has ever-growing wealth, but at

the same time ever-growing taxes that offset that wealth, so that the agent

is content with the consumption path defined by the economy’s real equi-

librium.4

A passive fiscal policy with φ1 > 0, therefore, guarantees that all con-

ditions for a private agent optimum are met on any of the paths for prices

and interest rates we have derived, including those with κ > 0. The infla-

tion rate (not just the price level) diverges to infinity on such a path, along

with the interest rate and velocity. So long as r is an increasing function

of v (ψ′′(v)v2 + 2vψ′(v) > 0), real balances shrink on these paths and, de-

pending on the specification of the ψ(v) function, may go to zero in finite

time.

With κ < 0, the initial interest rate and inflation rate are below the level

consistent with stable inflation and both the price level and the interest rate

decline on an exponential path. Since negative nominal interest rates are

not possible, it is impossible to maintain the Taylor rule when it prescribes,

as it eventually must on such a path, negative interest rates. The simplest

modification of the policy rule that accounts for this zero bound on the in-

terest rate, has r follow the right-hand side of (5) whenever this is positive

or r itself is positive, and otherwise sets r to zero. With this specification and

the passive fiscal rule (19) the economy has a second steady state (assuming

φ1 large enough to stabilize b), at r = 0, b = φ0/(φ1 − ρ− x). In this steady

state inflation is constant at −ρ− x. This steady state is stable.

4Note that, because the realized real rate of return on debt exceeds that on real assets F,

the properly discounted present value of future taxes exceeds the real value of debt on a

path with x > 0, and may even be infinite. “Ricardian” fiscal policy does not guarantee a

match between the present value of future taxes and the current real value of debt on this

non-rational-expectations path for the economy.


At this point we have approximately matched the model and conclu-

sions of Benhabib, Schmitt-Grohé, and Uribe (2001): This policy configura-

tion produces a pair of equilibria, with only one globally stable. Because the

equilibria with κ > 0 cannot be ruled out, and because there are many paths

for the economy that converge to the stable r = 0 point, the price level is

indeterminate.

However, the indeterminacy can be eliminated if we specify plausible

modifications of the policy rules for very high and low inflation rates. The

passive fiscal rule, when r and inflation are on an upward-explosive path,

requires that the conventional deficit, which includes interest expense, ex-

plode upward asymptotically at the same rate as r. This is required because

inflation is tending to reduce the real value of the debt, so large amounts of

additional nominal debt must be sold to the public to keep the real value of

the debt on its path converging to φ0/(φ1− ρ− x). This behavior of the pol-

icy authorities is implausible. It is natural to suppose that at very high infla-

tion rates the fiscal authorities would try to restrain the conventional deficit

by increasing τ, and that the monetary authorities would try to refrain from

exacerbating the conventional deficit by continuing to push r upward. In

Sims (forthcoming) it is shown in a model with only interest-bearing debt,

no currency or transactions costs, that even a tiny positive coefficient on in-

flation in the fiscal policy rule would make the explosive paths for inflation

unsustainable. In such a model it is also straightforward to describe policies

that differ from the standard passive active money, passive fiscal rule only

far from steady state and that also deliver a unique equilibrium price level.

In a model like that we use here, with transactions costs and non-interest-

bearing currency, the details of a modest policy shift at high or low inflation

rates that would guarantee a unique equilibrium price level depend on the

way transactions costs behave at very high and very low levels of veloc-

ity. Though the details might be complex, the essence of such a backup


policy is simple. If inflation gets too high, a modest fiscal and monetary

reform is undertaken that “punishes” market participants who have been

expecting ever-accelerating inflation by suddenly, but moderately, increas-

ing the value of the currency. If market participants see from the start that

this will happen, the explosive equilibrium path can never get started. If

market participants are not so far-sighted, the economy might start down

such a path, but as soon as people realized where the economy was headed

market forces would restore the stable-price equilibrium.

To eliminate the solution paths that converge to the low-inflation steady

state, we can invoke a different sort of realistic modification of the simple

active-money, passive-fiscal policy rules. The version of passive fiscal pol-

icy in Benhabib, Schmitt-Grohé, and Uribe (2001) makes the primary sur-

plus respond positively to the real value of all government liabilities, both

interest-bearing and zero-interest. But this makes little sense. There is no

need for taxes to increase with the real value of currency. In our model,

as the economy approaches the zero lower bound on nominal interest rates

(which it does in finite time on these deflationary paths if the standard Tay-

lor rule remains in place), real balances increase without bound. (v → 0 as

r → 0 and C does not decrease.) So long as these increased real balances

are not offset by correspondingly increased taxation and correspondingly

increased net lending by the government, they make the market value of

private wealth increase without bound in finite time. This violates the pri-

vate sector transversality condition. Less technically, individuals will not

be content with consumption satisfying the economy’s resource constraint

if the market value of their wealth grows arbitrarily large.

We can conclude that there is no internal contradiction in the conven-

tional practice of treating the price level as uniquely determined in models

with a Taylor rule. The justification for doing so, though, must appeal to a


backstop fiscal policy commitment — to tighten fiscal policy if inflation be-

comes too high, and to allow the real value of currency to increase without

bound, without raising taxes, if deflation takes hold. Central banks should

therefore not be structured to have no institutional link to the treasury, and

central bankers should not suggest in their public statements that they can

control inflation regardless of fiscal policy.

IV. FOUR LEVELS OF CENTRAL BANK BALANCE SHEET PROBLEMS

So far, we have said nothing about the central bank balance sheet, but

with the solution path for the economy in hand, assessing the time path of

the balance sheet is straightforward. The most severe problem, which we

can call level 4, is simply the possible indeterminacy of the price level. To

put this in the language of the central bank balance sheet, this is the point

that the central bank’s assets consist of the market value of its assets and its

potential seigniorage, both of which are valueless if currency is valueless.

But if it holds nominal debt as assets and issues reserves and currency as

liabilities, the central bank has no lever to guarantee the real value of either

side of its balance sheet. If the public were to cease to accept currency in

payment, it would become valueless, as would both sides of the central

bank balance sheet. That this cannot happen, either suddenly or as the end

point of a dynamic process, depends on fiscal commitments beyond the

central bank’s control.

The fiscal backing required for price level determinacy seems quite plau-

sible in the US. In Europe, because fiscal responsibility for the Euro is di-

vided among many countries that seem bent on frequently increasing doubts

about their ability to cooperate on fiscal matters, the possibility of a break-

down of the value of the Euro from this source cannot be entirely ruled out.

The next level of possible problem, level 3, arises because the notion

of determinacy via a backstop fiscal commitment assumes that the central


bank could maintain its commitment to an active policy rule during an in-

flationary excursion from the unique stable price path, up to the point that

fiscal backing is triggered. If we think of a unified government budget con-

straint and jointly determined monetary and fiscal policy, this is not an is-

sue. But if the central bank is concerned to maintain its policies without

requiring a direct capital injection from the treasury, or possibly even with-

out ever having to set its seigniorage payments to the treasury to zero, then

this could be a problem. And of course if markets perceive that the central

bank will abandon its policy rule to avoid having to seek treasury support,

this undercuts the argument for price determinacy. Showing formally how

these issues arise requires solving the model for time varying paths of inter-

est rates and velocity, so it is postponed to later sections of the paper.

If the market value of the assets of the central bank fall to a value be-

low that of their interest-bearing liabilities, it is possible that adherence to

the bank’s policy rule is impossible without a direct injection of capital.

This is only a possibility, however, because the bank has an implicit asset

in its future seigniorage. Even with assets below interest-bearing liabili-

ties at market value, the bank may be able to meet all its interest-paying

obligations and to restore the asset side of its balance sheet through accu-

mulation of seigniorage. Whether it can do so depends on its policy rule

and on the interest-elasticity of demand for currency (or more generally, for

its non-interest-bearing liabilities). This issue, of whether the central bank

might require a capital injection to maintain adherence to its policy rule in

a determinate-price-level equilibrium, is a level 2 balance sheet problem.

Finally, the central bank may be solvent in the sense that with the ex-

isting policy rule its assets at market value plus future seigniorage exceed

its total liabilities, yet following standard accounting rules and rules for de-

termining how much seigniorage revenue is sent to the treasury each pe-

riod may lead to episodes of zero seigniorage payments to the treasury.


Extended episodes of this type might be thought to raise issues of politi-

cal economy, if they led to public criticism of the central bank or to calls for

revising its governance.

V. INFLATION SCARE IN THE SIMPLE MODEL

Our first numerical example uses this simple model to compare a steady

state with ρ = β = ρ = .01 and x = 0 to one in which x jumps up to .02 at

time 0. This is an “inflation scare” scenario. The 2% per year inflation scare

shock produces a much larger increase in the nominal interest rate, because

the increased inflation expectations shrink demand for money and thereby

produce inflation, which prompts the central bank to raise rates further. If

the duration of the nominal assets on the central bank’s balance sheet is

positive, the permanent rise in rates reduces the time 0 market value of the

central bank’s assets. The simple model treats the debt as of maturity 0, but

this has no consequence except for the initial date capital losses, because for

t > 0 the perfect-foresight path requires that long and short debt has the

same time path of returns. We show two cases: initial assets of the central

bank A0 are three times the amount of currency outstanding or six times

the amount of currency outstanding with the initial deposit liabilities V0

plus currency matching A0 in each case.

The nominal capital losses, as a proportion of the new value of the assets,

are shown in Table 1. There cannot be any “level 2” problem for the central

bank unless the interest increase pushes its initial assets A below V. That

is, it not only has to have assets less than liabilities V + M, where M is

currency, it has to have A < V in order for a level 2 problem to arise. The

rise in interest rate reduces the demand for M, which has to be met either

by an decrease in A through open market sales or an increase in V. This

will dampen the effect on V − A of the rate rise. The value of V − B is

shown as the “gap” line in Table 1. Whether a level 2 problem actually arises


then depends on the discounted present value of the seigniorage after the

initial date, shown as “dpvs” in the table. For this example, even though

the gap between V and B gets quite large if we assume long durations for

the assets, the gap exceeds the discounted present value of seigniorage only

for durations of 10 years or more and for the (unrealistically large) balance

sheet with A0 six times outstanding currency.

This example should make it clear that the central bank can suffer very

substantial capital losses without needing direct recapitalization. On the

other hand, it shows that there are drawbacks to extreme expansion of cen-

tral bank holdings of long-maturity debt — an expanded balance sheet in-

creases the probability that interest rate changes could require a direct cap-

ital injection.

This simple model has omitted two sources of seigniorage, population

growth and technical progress, and has considered only a single, stylized,

shock to the balance sheet. We now expand the model to include these ex-

tra elements and calibrate the parameters and the nature of the shocks more

carefully to the situation of the US Federal Reserve. Of course our ability to

calibrate is limited by the sensitivity of results to the transactions cost func-

tion. We have little relevant historical experience with currency demand

at low or very high interest rates. Rates were very low in the 1930’s and

the early 1950’s, but the technology for making non-currency transactions

is very different now. It is difficult to predict how much and how fast people

would shift toward, say, interest-bearing pre-loaded cash cards as currency

replacements if interest rates increased to historically normal levels. We can

at best show ranges of plausible results.


VI. THE MODEL

The model borrows from Sims (2005). The household planner (whose

utility includes that of offspring, see Barro and Sala-i Martin (2004)) maxi-

mizes: ∫ ∞

0e−(β−n)t log(Ct)dt (22)

where Ct is per capita consumption, β is the discount rate, and n is popula-

tion growth, subject to the budget constraint:

Ct(1 + ψ(vt)) + Ft +Vt + Mt + qtBP

Pt=

Yeγt + (ρt − n)Ft + (rt − n)Vt

Pt+ (χ + δ− qtδ− n)

BP

P− n

Mt

Pt− τt. (23)

We express all variables in per-capita terms and initial population is normal-

ized to one. Ft and BPt are foreign assets and long-term government bonds

in the hand of the public, respectively, Vt denotes central bank reserves, Mt

is currency, τt is lump-sum taxes, Y is an exogenous income stream growing

at rate γ. Foreign assets and central bank reserves pay an exogenous real

return ρ and a nominal return rt, respectively. Long term bonds are mod-

eled as in Woodford (2001). They are assumed to depreciate at rate δ (δ−1

captures the bonds average maturity) and pay a nominal coupon χ + δ.5

The government is divided into two distinct agencies called “central

bank” and “fiscal authority”. The central bank’s budget constraint is(qt

BC

Pt− Vt + Mt

Pt

)ent

=

((χ + δ− δqt − nqt)

BC

P− (rt − n)

Vt

Pt+ n

Mt

Pt− τC

t

)ent. (24)

where BCt are long-term government bonds owned by the central bank, and

τCt are remittances from the central bank to the fiscal authority. The central

5We write the coupon as χ + δ so that at steady state if χ equals the short term rate the

bonds sell at par.


bank is assumed to follow the rule (5) for setting rt, the interest on reserves.

The central bank is also assumed to follow a rule for remittances, which

embodies two principles: i) remittances cannot be negative, ii) whenever

positive, remittances are such that the central bank capital (assets minus li-

abilities) remains constant in nominal terms over time (Hall and Reis (2013)

use a similar rule), that is:(qtBC

t −Vt −Mt

)ent = constant. (25)

Differentiating the condition above and plugging the resulting expression

into the central bank’s budget constraint, one can see that the two principles

result in the following rule for remittances:

τCt = max

{0, (χ + (1− qt)δ + qt)

BCt

Pt− rt

Vt

Pt

}. (26)

Solving the central bank’s budget constraint forward we can obtain its in-

tertemporal budget constraint:

qBC

0P0− V0

P0+∫ ∞

0(

Mt

Mt+ n)

Mt

Pte−∫ t

0 (ρs−n)dsdt =∫ ∞

0τC

t e−∫ t

0 (ρs−n)dsdt. (27)

Equation (27) shows that, regardless of the rule for remittances, their dis-

counted present value∫ ∞

0τC

t e−∫ t

0 (ρs−n)dsdt has to equal its left hand side,

namely the market value of assets minus reserves plus the discounted present

value of seigniorage∫ ∞

0(

Mt

Mt+ n)

Mt

Pte−∫ t

0 (ρs−n)dsdt. We can also compute

the constant level of remittances τCeγt (taking productivity growth into ac-

count) that satisfies expression (27).

τC =

(∫ ∞

0e(γ+n)t−

∫ t0 ρsdsdt

)−1

(q

Bc

P− V

P+∫ ∞

0(

Mt

Mt+ n)

Mt

Pte−∫ t

0 (ρs−n)dsdt)

. (28)

We also need to specify the central bank’s policy in terms of the asset

side of its balance sheet BCt . We assume that these follow some exogenous

process BCt = BC

t . Government debt is assumed to be held either by the


central bank or the public: Bt = BCt + BP

t . The budget constraint of the fiscal

authority is(Gt − τt + (χ + δ− δqt − nqt)

BP

)ent =

(τC

t + qtBPt

)ent, (29)

where Gt is government spending. The rule for τt is given by:

τt = ξ0eγt + (ξ1 + n + γ)

(q

BP

P+

VP

). (30)

This rule makes the debt to GDP ratio bt =

(q

BP

P+

VP

)e−γt converge as

long as ξ1 > β− n. The initial level of foreign assets in the hand of the pub-

lic, central bank reserves, and currency are FP0 , V0, and M0, respectively.

As in the simple model the first order condition for the household’s

problem with respect to C, FP, B, V, and M yield the Euler equation (12), the

Fisher equation (10),6 the money demand equation (11), and the arbitrage

condition between reserves and long-term bonds:

χ + δ

q− δ +

qq= r. (31)

The solutions for r is given by equation (14), and those for inflationPP

and

velocity v follow from equations (10) and (11), respectively. The growth rate

of consumptionCC

, is given by

CC

= (ρ− β)− 2ψ′(v) + vψ′′(v)1 + ψ(v) + vψ′(v)

v, (32)

which obtains from differentiating expression (12). Differentiating the def-

inition of velocity (3) we obtain an expression for the growth rate of cur-

rency:MM

=PP+

CC− v

v. (33)

6Note that short term debt was called B in the simple model, and was issued by the

fiscal authority. Here it is called V, and is issued by the central bank.


The economy’s resource constraint is given by

C(1 + ψ(v)) + F = (Y− G)eγt + (ρ− n)F, (34)

where F = FP + FC is the aggregate amount of foreign assets held in the

economy (we assume that the central banks foreign reserves FC are zero),

and where we assumed Gt = Geγt. Solving this equation forward we obtain

a solution for consumption in the initial period:

C0

(∫ ∞

0(1 + ψ(v))e−

∫ t0 (ρs− C

C−n)dsdt)= F0 + (Y− g)

∫ ∞

0e(γ+n)t−

∫ t0 ρsdsdt,

(35)

Given velocity v and the level of consumption, we can compute real money

balancesMP

, the initial price level P0, and seigniorageMP

+nMP

=

(MM

+ n)

MP

(using (33)), and the present discounted value of seigniorage

∫ ∞

0

(MM

+ n)

MP

e−∫ t

0 (ρs−n)dsdt = c0

∫ ∞

0

(MM

+ n)

v−1e−∫ t

0 (ρs− CC−n)dsdt.

Finally, solving (31) forward we find the current nominal value of long-term

bonds

q0 = (χ + δ)∫ ∞

0e−(∫ t

0 rsds+δt)

dt. (36)

VI.1. Steady state. At a steady state where ρ = β + γ, r = ρ + π, v satis-

fies v2ψ′(v) = rss. Steady state consumption is given by Ct = C0eγt where

C0 =(β− n)F0 + Y− G

1 + ψ(v), and real money balances are given by

MP ss =

C0

veγt. seigniorage is given by (π + γ + n)

C0

ve(γ+n)t and its present dis-

counted value is given by (π + γ + n)C0

v(β− n).

VI.2. Central bank’s solvency, accounting, and the rule for remittances.

For some of the papers discussed in the introduction the issue of central

bank’s solvency is simply not taken into consideration: the worst that can

happen is that the fiscal authority may face an uneven path of remittances,

with possibly no remittances at all for an extended period. We acknowledge


the possibility that remittances may have to be negative, at least at some

point. This is what we mean by solvency.

Like Bassetto and Messer (2013), we approach the issue of central bank’s

solvency from a present discounted value perspective. If the left hand side

of equation (27) is negative, the central bank cannot face its obligations, i.e.,

pay back reserves, without the support of the fiscal authority. An inter-

esting aspect of equation (27) is that its left hand side does not depend on

many of aspects of central bank policy that are recurrent in debates about

the fiscal consequences of central bank’s balance sheet policy. For instance,

the future path of BCt does not enter this equation: whether the central bank

holds its assets to maturity or not, for instance, is irrelevant from an ex-

pected present value perspective. Intuitively, the current price qt contains

all relevant information about the future income from the asset relative to

the opportunity cost rt. Whether the central bank decides to sell the assets

and realize gains or losses, or keep the assets in its portfolio and finance

it via reserves, does not matter. Similarly, whether the central bank incurs

negative income in any given period, and accumulates a “deferred asset”,

is irrelevant from the perspective of the overall present discounted value of

resources transferred to the fiscal authority.7 In fact, we will see later that in

some cases scenarios associated with higher remittances in terms of present

value are also associated with a deferred asset.

Finally, the issue of “remittances smoothing” is also, from a purely eco-

nomic point of view, a non issue. In perfect foresight the central bank

can always choose a perfectly smooth path of remittances (in fact, this is

τCt = τCeγt). In a stochastic environment Barro’s results on tax smooth-

ing would apply: remittances would move with innovations to the the left7As we will see later central bank accounting does not let negative income affect capital.

The budget constraint (24) implies however that negative income results in either more

liabilities or less assets. In order to maintain capital intact, a deferred asset is therefore

created on the asset side of the balance sheet.


hand side of equation (27). But there are accounting rules governing central

banks’ remittances. Hence these may not be smooth and may depend on

the central bank’s actions, such as holding the assets to maturity or not. We

recognize that the timing of remittances can matter for a variety of reasons:

tax smoothing, political pressures on the central bank, et cetera. For this

reason we assume a specific rule for remittances that very loosely matches

those adopted by actual central banks and compute simulated paths of re-

mittances under different assumptions. Appendix A discusses this rule.

VI.3. Functional forms and parameters. Table 2 shows the model parame-

ters. We normalize Y − g to be equal to 1, and set F0 to 0.8 Since we do not

have investment in our model, and F0 = 0, Y− G in the model corresponds

to national income Y minus government spending G in the data (data are

from Haver analytics, mnemonics are Y@USNA and G@USNA, respectively).

All real quantities discussed in the remainder of the paper should therefore

be understood as multiples of Y− G, and their data counterparts are going

to be expressed as a fraction of national income minus government spend-

ing ($ 11492 bn in 2013Q3). Our t = 0 corresponds to the beginning of 2014.

We therefore measure our starting values for the face value of central bank

assetsBC

P, reserves

VP

, and currencyMP

using the January 3, 2014 H.4.1 re-

port (http://www.federalreserve.gov/releases/h41/), which mea-

sures the Security Open Market Account (SOMA) assets.9 The model pa-

rameters are chosen as follows. The discount rate β, productivity growth γ,

8Note from the steady state calculations that we could choose F0 6= 0 and use instead

the normalization (β − n)F0 + y − g=1, hence setting F0 6= 0 simply implies a different

normalization.9The January 3, 2014 H.4.1 reports the face value of Treasury ($ 2208.791 bn ), GSE debt

securities ($ 57.221 bn), and Federal Agency and GSE MBS ($ 1490.160 bn), implying that

BC0 is $ 3756.172 bn, the value of reserves V (deposits of depository institutions, $ 2374.633

bn) and currency M (Federal Reserve notes outstanding, net, $ 1194.969 bn).


and population growth n are 1 percent, 1 percent, and .75 percent, respec-

tively. These values are consistent with Carpenter et al.’s assumptions of a

2% steady state real rate.

The policy rule has inflation and interest rate smoothing coefficients θπ

and θr of 2 and 1, respectively, which are roughly consistent with those of in-

terest feedback rules in estimated DSGE models (e.g., Del Negro, Schorfheide,

Smets, and Wouters (2007); note that θr = 1 corresponds to an interest rate

smoothing coefficient of .78 for a policy rule estimated with quarterly data).

The inflation target θπ is 2 percent (hence θ0 = β + γ− (θπ − 1)πss=-.0025).

We use the functional form

ψ(v) = ψ0v

1 + ψ1v(37)

for the transaction costs, with ψ0 = 2× 10−6 and ψ1 = −0.055. Figure 1

shows the scatter plot of quarterlyMPC

= v−1 and the annualized 3-month

TBill rate in the data (where M is currency and PC is measured by nom-

inal PCE)10, where blue crosses are post-1980 data, and crosses are 1947-

1980 data (arguably less relevant). The black curve in figure 1 shows the

relationship between velocity and interest rates as implied by the model

(equation (11)). The parameters ψ0 and ψ1 are chosen to i) match currency

demand in real termsMP

in 2013Q4 at current rates (r0=.0025),11 ii) so thatMPC

= v−1 asymptotes at ψ1 = .055 when rates go to infinity, which implies

that model-implied velocity is roughly in line with the experience of the

early 1980s, as shown in figure 1. The implied transaction costs at steady

state are negligible - about .03 percent of Y-G. We will later also consider

10Data are from Haver, with mnemonics C@USNA, FMCN@USECON, and FTBS3@USECON

for PCE, currency, and the Tbill rate, respectively.11Matching inverse velocity

MPC

in 2013Q4 as opposed to real money demandMP

yields

very similar results.


alternative parameterizations of currency demand.12 Finally, we choose

χ – the average coupon on the central bank’s assets – to be 3.5 percent,

roughly in line with the numbers reported in figure 6 of Carpenter, Ihrig,

Klee, Quinn, and Boote (2013). Chart 17 of the April 2013 FRBNY report

on “Domestic Open Market Operations during 2012”13 shows an average

duration of 6 years for SOMA assets (SOMA is the System Open Market

Account, which represents the vast majority of the Federal Reserve balance

sheet). We therefore set 1/δ =6.

VII. SIMULATIONS

As a baseline simulation we choose a time-varying path of short term

nominal interest rates that roughly corresponds to the baseline interest path

in Carpenter, Ihrig, Klee, Quinn, and Boote (2013). We generate this path by

assuming that the real rate ρt remains at a low level ρ0 for a period of time

T0 equal to five years, and then reverts to the steady state ρ at the rate ϕ1:

ρt =

ρ0, for t ∈ [0, T0]

ρ + (ρ0 − ρ)e−ϕ1(t−T0), for t > T0.(38)

Given the path for ρt, equation (14) generates the path for the nominal short

term rate (we set κ = 0 for the baseline simulation). The baseline paths of

ρt, rt and inflation πt are shown as the solid black lines in the three panels

of Figure 2.

12We have performed non-linear least squares regression of equation (11) using the

interest rate and velocity data shown in figure 1. Some of the estimates of ψ0 and ψ1 –

particularly those using post-1980 data – are quite close to those reported in table 2. These

estimates generally yield a value of ψ1 close to −.06 in order to fit the high inflation data of

the early 1980s, which we as implying too large an asymptote forMPC

= v−1 if rates were

to become very large, in light of current transaction technology.13http://www.newyorkfed.org/markets/omo/omo2012.pdf.


Given the path for ρt and rt we can compute q and the amount of re-

sources, both in terms of marketable assets and present value of future

seigniorage, in the hands of the central bank. The first row of table 3 shows

the two components of the left hand side of equation (27), namely the mar-

ket value of assets minus reserves (column 1) and the discounted present

value of seigniorage∫ ∞

0(

MM

+ n)MP

e∫ t

0 (ρs−n)dsdt (column 2). The third col-

umn shows the sum of the two, which has to equal the discounted present

value of remittances∫ ∞

0τCe

∫ t0 (ρs−n)dsdt. Column 4 shows τC as defined in

equation (28): the constant level of remittances (accounting for the trend in

productivity) that would satisfy equation (27), expressed as a fraction of Y-

G like all other real variables.14 Last, in order to provide information about

how the numbers in column 1 are constructed, column 5 shows the nominal

price of long term bonds q at time 0.

Under the baseline the real value of the central bank’s assets minus lia-

bilities is 14.6 percent of Y-G – which is larger than the difference between

the par value of assets minus reserves reported in table 2 given that q is

above one under the baseline. Its value is 1.08, which is above the 1.04 ra-

tio of market over par value of assets reported in Federal Reserve System

(2014)15 The discounted present value of seigniorage is almost an order of

magnitude larger, however, at about 99 percent of Y-G, and represents the

bulk of the central bank resources (and therefore the present discounted

value of remittances), which are 114 percent of Y-G. The constant (in pro-

ductivity units) level of remittances τC that satisfies the present value rela-

tionship is .26 percent of Y-G, about $ 29 bn per year, quite lower than the

amount remitted for 2013 and 2012 according to Federal Reserve System

(2014) ($ 79.6 and $ 88.4 bn, respectively).

14That is, the amount τC such that τCt = τCeγt satisfies the present value relationship.

15Page 23 and 29 shows the par and market (fair) value of Treasury and GSE debt secu-

rities, and Federal Agency and GSE MBS, respectively.


The left and right panels of Figures 3 show inverse velocity M/PC and

seigniorage, expressed as a fraction of Y-G, in the data (1980-2013) and in

the model (under the baseline simulation), respectively. A comparison of

the two figures shows that the drop in M/PC as interest rates renormalize

under the baseline simulation (from about .085 to .065, left axis) is roughly

as large as the rise in M/PC as interest rates fell from 2008 to 2013. Partly

because the model may likely over-predict the fall in currency demand, and

more importantly because consumption declines (real interest rates are very

low at time 0, inducing unrealistic above trend consumption), seigniorage

falls to negative territory for roughly six years. After that, it converges to

almost .3 percent of Y-G, a level that is in the low range of the post-1980

observations. For both reasons the present discounted value of seigniorage

reported in table 2 for the baseline simulation is likely to be a fairly conser-

vative estimate.

Finally, the left panel of Figure 4 shows remittances (computed as de-

scribed in section A) under two scenarios for the path of assets BC: in the

first scenario (solid line) the central bank lets its assets depreciate, while in

the second one it actively sells assets at a rate of 20 percent per year. These

scenarios highlight the fact that different paths for the balance sheet can

imply different paths for remittances, even though their expected present

value remains the same (this is the dotted line in figure 4, which shows

τCeγt).

Next, we consider alternative simulations where the economy is subject

to different “shocks.” In each of these simulations all uncertainty is revealed

at time 0, at which point the private sector will change its consumption

and portfolio decisions and prices will adjust. We will use the subscript

0− to refer to the pre-shocks quantities and prices (that is, the time 0 quan-

tities and prices under the baseline simulation). For each simulation, Ta-

ble 3 will report the new market value of assets minus reserves in real term


(q0BC

0−

P0− V0

P0). By assumption the central bank will not change its assets BC

0−

after the new information is revealed, but the private sector will change its

time 0 currency holdings given that interest rates may have changed. This

necessarily leads to a change in reserves (given that central bank’s assets are

unchanged) equal toV0 −V0−

P0= −M0 −M0−

P0in real terms. We report this

quantity separately in column 5.

Last, for each scenario we also report the level of the balance sheet BC

such that, for any balance sheet size larger than BC, the present discounted

value of remittances (see equation (27)) becomes negative after the shock.

We refer to this situation as the central bank becoming “insolvent”, in the

sense that at some point it will need resources from the fiscal authority.

Specifically, assume the central bank expands its balance sheet by ∆BC at

time 0− (right before the shock takes place) by buying assets at price q0−

and pays with it by expanding reserves by an amount ∆V = q0−∆BC. How

large can ∆BC be to still satisfy

q0(

BC + ∆BC)−V − ∆VP0

+M0 −M0−

P0+∫ ∞

0(

Mt

Mt+ n)

Mt

Pte−∫ t

0 (ρs−n)dsdt ≥ 0 (39)

after the “shock”? We report B/BC = 1 +∆BC

BC , where BC is the 2013Q4

level of the balance sheet reported in table 2. Of course, the reason why

with a larger balance sheet the central bank may become insolvent is that q

is lower under the alternative simulations, and hence the central bank may

experience losses, in addition to possibly having less seigniorage in present

value.

The first alternative scenario we consider is a “higher rates” path similar

to one considered by Carpenter, Ihrig, Klee, Quinn, and Boote (2013). Under

this new path real rates converge to a 1 percent higher steady state, and so


will short term nominal rates given that the central bank inflation target has

not changed. We choose the new starting value for ρ, ρ0, so that the initial

rate remains at 25 basis points. The red solid lines in the three panels of

Figure 2 show the “Higher Rates” paths for the nominal and the real short

term rates and inflation, respectively. In these simulations we assume that

the central bank recognizes the change in the steady state ρ = β + γ, and

adjusts its Taylor rule coefficient r = ρ + π accordingly.

We consider two different reasons why the new steady state ρ is higher:

a higher discount rate β and a higher growth rate of technology γ. While

the new value of q is the same in both cases (the interest rate path is the

same), the present value of seigniorage shown in column 2, and therefore

the present value of remittances shown in column 3, is quite different. In

the high β case the current value of the future income from seigniorage falls

by almost one order of magnitude, as future seigniorage is discounted at

a higher real rate. In the high γ case the economy is growing faster, and

so does money demand and future seigniorage. As it turns out, in both

cases (higher β and higher γ) the level of τC is higher than in the baseline

case. This may seem surprising in the higher β case since the present value

of seigniorage is lower than under the baseline simulation. However, the

central bank is now earning a higher return on its assets, and can therefore

afford a higher level of remittances.16

We also consider the case where the central bank does not recognize that

the steady state real rate has increased, and therefore leaves r unchanged in

its reaction function. This is an extreme version of the more realistic case16Note that our conclusion is different from that reached in the analysis of Carpenter,

Ihrig, Klee, Quinn, and Boote (2013), which take seigniorage as given and focus on the

effect of the higher nominal interest rates on the value of the central bank’s assets qBC,

which falls following the drop in q. The effect of the higher real rate of return on future

central bank’s revenues and, especially in the high γ case, on future seigniorage, trumps in

our simulation the negative effect on q.


where the central bank adjusts its reaction function slowly to changes in

the real economy (see Orphanides (2002)). The red dash-and-dotted lines in

Figure 2 show the paths of nominal interest rates and inflation, respectively,

in this scenario. Inflation is higher relative to the case where the central

bank changes r (solid red lines) because the central bank’s reaction function

calls for rates that are too low. In equilibrium, since inflation is higher, rates

end up being also higher. As a consequence, q falls by more than in the case

where the central bank adjusts r (see the rows labeled “Higher rates/same

intercept” in Table 3). In addition, because of the higher interest rates the

private sector is no longer willing to hold as much currency and turns it into

reserves, and hence a higher fraction of the central bank’s liabilities becomes

interest bearing relative to the baseline scenario. Because of the higher in-

flation, however, given our assumptions about money demand seigniorage

is higher (compare rows 2 and 4 for the higher β case, and 3 and 5 for the

higher γ case, of Table 3). As a consequence, the fall in q does not raise any

solvency issues under these scenarios, which the central bank could have

withstood even with a balance sheet more than five times as large as the

actual one (see column 7). In fact, average yearly remittances τC increase by

about .10 percent of Y-G, roughly $ 10 bn.

The results are very sensitive to the inflation response in the policy re-

action function. The blue dash-and-dotted and dotted blue lines in the top

left panel of Figure 5 show the interest rate path corresponding to an infla-

tion coefficient θπ of 3 and 1.05.17 As is usually the case in stable rational

expectations equilibria, a higher inflation coefficient in the interest rate rule

induces a lower equilibrium response of inflation, and therefore a lower

equilibrium response of interest rates – and vice versa when the inflation

response is lower. In fact, we see that when θπ is 1.05, interest rates reach 25

17In these simulations we change the time 0 real rate so that under the baseline scenario

the nominal rate is still 25 basis points.


percent. Consequently, q falls to less than half its value under the baseline

scenario, and the market value of assets minus reserves q0BC

0−

P0− V0

P0falls to

negative levels (see row 11 of Table 3). The implication of this finding is that

under a large balance sheet the central bank may want to respond more ag-

gressively to inflation, if it is concerned about fluctuations in the values of

its assets.

Even in the θπ = 1.05 case central bank’s solvency is not an issue, how-

ever. The present value of seigniorage has a roughly fivefold increase rela-

tive to the θπ = 2 case, so in spite of the large decline in the value of assets,

the present value of remittances remains positive. In fact, the present value

of remittances would remain positive even if we assumed the central bank

balance sheet to be more than three times as large as the current one (col-

umn 7). We discuss later what would happen in this scenario under a less

favorable outlook for seigniorage (induced by a different parameterization

of the money demand function).

Next, we consider simulations where for a given period of time (10

years) the private sector is concerned about a sudden jump in the price level,

and therefore demands a premium x for holding nominal bonds. We call

these scenarios “inflation scares” and set x to 2 percent. The red solid lines

in the top right and bottom left panels of Figure 5 show what happens to

short term nominal interest rates and inflation, respectively, under this sce-

nario. Because of the higher inflation expectations the central bank, which

follows the interest rule, is forced to raise nominal rates over the baseline

path. This scenario has a number of effects on the central bank’s balance

sheet, as shown in row 6 of Table 3. The market value of assets minus re-

serves q0BC

0−

P0− V0

P0drops to less than half its baseline value, both because q

falls and because the private sector turns currency into reserves. However,

the present discounted value of seigniorage computed under our assump-

tions on money demand is slightly higher than under the baseline case. The


fact that seigniorage holds up, and that its present discounted value is large,

implies that the present value of resources in the hands of the central bank is

not much affected under the inflation scare simulations. As a consequence,

even with a much larger balances sheet the central bank could have with-

stood the fall in the value of its assets without ever needing any resources

from Fiscal Authority, at least in expectations. Also, because seigniorage

is still large the decrease in average remittances τC is small relative to the

baseline case, from .26 to .24 percent of Y-G.

The blue lines in the top right and bottom left panels of Figures 5 show

the responses of interest rates and inflation, respectively, under policy rules

different from the baseline for this scenario. As in the “Higher rates/same

intercept” case, if policy reacts more aggressively to inflation than under the

baseline in equilibrium inflation and short term interest rise less, and vice

versa if policy reacts less aggressively to inflation. Rows 9 and 12 of Table 3

document the effect of these alternative policies on the central bank’s bal-

ance sheet and show that, again not surprisingly, the effect of the “inflation

scare” simulation on q is stronger the lower the response to inflation θπ.

In fact, under the lower θπ policy, the market value of assets and reserves

becomes negative. The central bank’s overall resources (column 3) are still

sizable, and average remittances are the same as under the θπ = 2 policy.

Again, this is because the higher inflation experienced under the lower θπ

policy yields greater seigniorage (column 2).

Finally, we consider explosive paths where κ in equation (14) is different

from zero. The solid red line in the bottom right panel of Figure 5 shows one

of these paths (with κ = 10−4) under the baseline policy response. Given

the rise in rt under this scenario, q drops by .2 relative to the baseline (row

7 of Table 3). However, q0BC

0−

P0− V0

P0increases to almost four times Y-G.

The reason for this counterintuitive result is that the under our assumption


for money demand transaction costs explode as rates reach infinity. There-

fore agents will front-load consumption as much as possible while interest

rates and transaction costs are still low, and this will drive up money de-

mand. Moreover, under these explosive paths seigniorage will reach very

high levels because under our parametrization M/Pc (inverse velocity) has

a lower bound, and the public continues to be willing to hold currency no

matter how high rates are. The present discounted value of seigniorage will

therefore be very high (we can show that since seigniorage asymptotes to

a finite level, the present discounted value is actually a finite number, but

is so large that our integration routine does not converge, hence we report

“Inf” in Table 3).

The dash-and-dotted and dotted blue lines in the bottom right panel

of Figure 5 show the responses under different θπ coefficients. In the case

of unstable solutions, the inflation response coefficient in the interest rule

plays the opposite role relative to the stable solution case (see Cochrane

(2011)): the stronger the response, the faster inflation and interest rates ex-

plode. The market value of central bank’s assets q surely falls more with a

higher θπ, but as in the θπ = 2 case the increase in the demand for money

in the initial period and the subsequent large seigniorage imply that central

bank’s solvency is never in question under these paths – quite the contrary,

the central bank transfers very large resources to the fiscal authority.

VII.1. Calibration with money demand equal to zero for r > 100%. We

have seen that under many scenarios where the value of the central bank’s

assets drops substantially, seigniorage can save the day in terms of central

bank’s solvency. This conclusion is not robust to the assumptions about

money demand, however. Table 4 show the central bank’s resources un-

der the baseline, higher rates/same intercept, inflation scare, and explosive

paths simulations for a money demand function that drops to zero when

interest rates are above 100 percent (the other parameter of the transaction


cost function (37) is still chosen so that currency holdings match 2013Q4

holdings for r = .0025). As shown in the left panel Figure 6, this money de-

mand function predicts an unrealistically large drop in money demand as

interest rates return to steady state, as well as a too low steady state level of

seigniorage. Under the baseline simulation, the present value of seigniorage

is therefore below a third of its value under the benchmark money demand

function. But it does have the arguably realistic feature that for very high

rates of inflation the private sector will use means of transactions other than

cash. As a consequence under the explosive path there is no time 0 jump

in money demand (people are no longer twisting their consumption pro-

file because of transaction costs) and no dramatic increase in seigniorage to

cushion the effect of the fall in q. Seigniorage is actually negative in present

value under the explosive paths, as the public dumps its currency holdings

(row 12 of Table 4). Under explosive paths with κ ≥ 10−4 the central bank,

therefore, becomes insolvent: it can no longer stick to the Taylor rule and

at the same time honor its liabilities without an intervention from the fiscal

authority.

VIII. SELF-FULFILLING SOLVENCY CRISES

As we have already observed, a central bank cannot guarantee determi-

nacy of the price level in the absence of fiscal backing. Our detailed sce-

narios in the previous sections have all assumed (except in the κ > 0 cases)

that this backing was present. But even when the backing is present, a cen-

tral bank that is firmly committed to not accepting (or incapable of drawing

on) fiscal support, in the sense of capital injections from the treasury, can

create indeterminacy in the price level. The problem is that commitment

to a policy rule that stabilizes inflation, like a Taylor rule with large coeffi-

cient on inflation, may under certain conditions require a capital injection


from the treasury to be sustainable. If the central bank, to avoid the cap-

ital injection, switches policy so as to generate more seigniorage, multiple

non-explosive equilibria can arise. We give examples of this possibility in

this section. In our calibrated version of the U.S. economy the particular

form of multiple equilibria we consider would only arise for levels of the

central bank’s balance sheet larger than the current one, although this result

depends crucially on the properties of the demand for currency.

What if the public believes that, were the central bank to face the issue of

solvency, it would resort to seigniorage creation? Entertaining the possibil-

ity of central bank’s insolvency would then lead the public to expect higher

future inflation and nominal interest rates. These expectations would re-

sult in a lower value of long term nominal assets today, so that the central

bank’s assets qBC could become worth less than its interest bearing liabil-

ities V. If the current present discounted value of seigniorage is not large

enough to cover this gap, the central bank may have to resort to raising

more seigniorage, thereby validating the initial belief. The larger is the size

of the central bank’s balance sheet, and the longer its duration, the larger

is the gap in qBC −V that would arise because of future expected inflation,

and the likelihood of these alternative equilibria.

If there is the possibility of indeterminacy, these multiple equilibria can

take many forms. We focus on a particular type of multiple equilibria,

where agents expect that at time t = T the central bank will change its infla-

tion target to π for a period ∆, and revert to the old rule with inflation target

π afterwards (for t > T + ∆). The appropriately modified version of equa-

tion (14) provides the solution for the future path of interest rates. Given

the path for rt we can solve for all other endogenous variables exactly as in

the model above. We can in particular obtain, under this alternative equilib-

rium, the value of long term assets q0(π), the initial price level P0(π), and

the present discounted value of seigniorage in real terms at time 0, which


we can call PDVS0(π) =∫ ∞

0(

Mt

Mt+ n)

Mt

Pte−∫ t

0 (ρs−n)dsdt. All of these ob-

jects will be a function of π (and of T and ∆ as well). For each T and ∆,

we can then find what expected future inflation π needs to be to generate a

solvency crisis, that is, we can look for solutions of

q0(π)BC0− −V0

P0(π)+ PDVS0(π) = 0, (40)

if any exist. These solutions are possible self-fulfilling solvency crises, in

the sense that the expectation that the central bank will switch to a new rule

with target π will produce a gap in the value of central bank’s assets minus

liabilitiesq0(π)BC

0− −V0

P0(π)that will have to be filled with future seigniorage

PDVS0(π). In order to generate this future seigniorage the central bank

will have to validate the public expectations and switch temporarily to the

rule with higher inflation target.

Using our simple calibrated model we search for these alternative equi-

libria. In particular, for given π, T, and ∆ we find the minimum level of the

balance sheet BC for which equation (40) has a solution. The top panel left

panel of figure 8 shows this minimum balance sheet level (relative to the

current level) as a function of inflation in there alternative regime (π ), and

the duration of the alternative regime (∆). To repeat, these are the balance

sheet levels for which the solvency constraint would become binding if the

public expects a regime with inflation π and duration ∆ to materialize in

T = 3 years.

The figure shows that under the baseline money demand calibration,

these threshold balance sheet limits are much larger than the current one.

The left middle and bottom panels of figure 8 explain why this is the case.

The middle panel shows what happens toq0(π)BC

0− −V0

P0(π)in the alterna-

tive equilibrium under the current balance sheet size. The figure shows that

for large enough π and duration ∆ the real value of assets minus interest


bearing liabilities does become significantly negative. The bottom panel

of figure 8 show that for these values the level of seigniorage PDVS0(π)

overshadows this balance sheet loss, however. Hence the results in the top

panel: the size of the balance sheet would have to be much larger than the

current one for the balance sheet loss to be of the same size of the increase in

seigniorage. In other words, under the current level of the balance sheet, the

type of alternative equilibria we consider cannot arise because the increase

in seigniorage triggered by the temporary higher inflation regime is larger

than the balance sheet loss caused by the fall in q.18 The Laffer curve in the

left panel of figure 7 shows why the increase in seigniorage is so large un-

der the baseline parameterization of money demand: even for large interest

rates money demand asymptotes to positive values, and seigniorage grows

linearly with inflation.19

Under the alternative money demand the threshold balance sheet lim-

its are much closer to one, as a consequence of the fact that the increase in

seigniorage is smaller. The right panel of figure 7 shows that seigniorage

peaks at about 40 percent steady state inflation, and becomes zero for in-

flation larger than one hundred percent. Hence seigniorage increases much

less with π than in the baseline case, as shown in the bottom right panel of

figure 8 (since the duration of the alternative regime is relatively short we

find that seigniorage still increases for values of π up to 500 percent). Still,

we find that the threshold balance sheet levels that would generate multi-

plicity are all larger than one for the (π, ∆π) values shown here. Therefore,

even for the alternative money demand multiple equilibria under the cur-

rent size of the balance sheet could be possible only if the public expected

very large inflation and/or very long duration of the alternative regime. In-

deed, we find that that for instance π = 5, that is, 500 percent inflation, for

18We searched for alternative values of T as well, and the results are not very different.19One caveat to these simulations is that we maintain the hypothesis of passive fiscal

policy even under high inflation rates, which may not be realistic.


158 periods would be an equilibrium under the current level of the balance

sheet. These results are in line with what shown in the previous sections,

namely, one has to assume fairly radical scenarios for solvency to become

an issue.

IX. CONCLUSIONS

To Be Written

REFERENCES

BARRO, R. J., AND X. SALA-I MARTIN (2004): Economic Growth. The MIT

Press, second edn.

BASSETTO, M., AND T. MESSER (2013): “Fiscal Consequences of Paying In-

terest on Reserves,” Federal Reserve Bank of Chicago working paper No. 2013-

04.

BENHABIB, J., S. SCHMITT-GROHÉ, AND M. URIBE (2001): “The Perils of

Taylor Rules,” Journal of Economic Theory, 96, 40–69.

CARPENTER, S. B., J. E. IHRIG, E. C. KLEE, D. W. QUINN, AND A. H.

BOOTE (2013): “The Federal Reserve’s Balance Sheet and Earnings: A

primer and projections,” FEDS Working Paper, 2013-01.

CHRISTENSEN, J. H. E., J. A. LOPEZ, AND G. D. RUDEBUSCH (2013): “A

Probability-Based Stress Test of Federal Reserve Assets and Income,” Fed-

eral Reserve Bank of San Francisco Working Paper 2013-38.

COCHRANE, J. H. (2011): “Determinacy and Identification with Taylor

Rules,” Journal of Political Economy, 119(3), 565–615.

CORSETTI, G., AND L. DEDOLA (2012): “The mystery of the printing press:

self-fulfilling debt crises and monetary sovereignty,” mimeo Oxford Uni-

versity.


DEL NEGRO, M., F. SCHORFHEIDE, F. SMETS, AND R. WOUTERS (2007):

“On the Fit of New Keynesian Models,” Journal of Business and Economic

Statistics, 25(2), 123 – 162.

FEDERAL RESERVE SYSTEM (2014): “The Federal Reserve Banks. Com-

bined Financial Statements as of and for the Years Ended De-

cember 31, 2013 and 2012 and Independent Auditors’ Report,”

http://www.federalreserve.gov/monetarypolicy/files/BSTcombinedfinstmt2013.pdf.

HALL, R. E., AND R. REIS (2013): “Maintaining Central-Bank Solvency un-

der New-Style Central Banking,” manuscript.

ORPHANIDES, A. (2002): “Monetary-Policy Rules and the Great Inflation,”

The American Economic Review, 92(2), 115–120.

SIMS, C. A. (2005): “Limits to Inflation Targeting,” in The Inflation-Targeting

Debate, ed. by B. S. Bernanke, and M. Woodford, vol. 32, chap. 7, pp. 283–

310. NBER Studies in Business Cycles.

(forthcoming): “Paper Money,” American Economeic Review, 103(2),

1–24.

WOODFORD, M. (2001): “Fiscal Requirements for Price Stability,” Journal of

Money, Credit and Banking,, 33(1), 669–728.

APPENDIX A. RULE FOR REMITTANCES

The central bank is assumed to follow a rule for remittances, which em-

bodies two principles: i) remittances cannot be negative, ii) whenever posi-

tive, remittances are such that the central bank capital measured at historical

costs remains constant in nominal terms over time, that is:20

K =(

qBC −V −M)

ent = constant. (41)

The historical price q evolves according to

˙q = (q− q)max{

0,BC

BC + δ + n}

(42)

20Hall and Reis (2013) use a similar rule, but measure capital at market prices.


where the max operator is there because q changes only if the central bank

is acquiring assets (recall that bonds depreciate at a rate δ and that BC is

defined in per capita terms, so that BC = −(δ+ n)BC implies that the central

bank is letting its assets mature). Differentiating condition (41) above and

using the central bank’s budget constraint (24), one obtains a condition for

nominal remittances:

PτC = (χ− δ(q− 1)) BC +

(˙q− (q− q)

(BC

BC + δ + n))

BC − rV. (43)

This condition resembles closely the accounting practice of central banks.

The first term, (χ− δ(q− 1)) BC, measures coupon income χ net of the

amortization of historical costs δ(q − 1), times the par value of bonds BC.

The second term equals the realized gains/losses,−(q− q)(

BC

BC + (δ + n))

BC,

from assets sales (that is,BC

BC ≤ −(δ + n)), since in this case ˙q = 0. When

the central bank is acquiring assets (BC

BC > −(δ + n)) this second term is

zero because ˙q and (q − q)(

BC

BC + δ + n)

cancel each other out. The last

term, rV, captures the interest paid on reserves. Whenever net income (the

left hand side of equation 43) is negative, the central bank would have to

extract negative remittances from the fiscal authority to keep its capital con-

stant. If it cannot do this, its capital declines. Whenever internal accounting

rules prevent capital from declining a deferred asset is created. Remittances

remain at zero until this deferred asset is extinguished (i.e., capital is back

at the original level). Hence our rule for remittances is

τCt = max

{0, (χ− δ(q− 1))

BC

P

+

(˙q− (q− q)

(BC

BC + (δ + n))

BC

P

)− r

VP

}I{K≥K0}, (44)


where I{K≥K0} is an indicator function equal to one only if current capital K

is at least as large as initial capital K0 (that is, the deferred asset has been ex-

tinguished). In practice central bank’s capital will not be constant over time,

but will likely grow along with nominal income. This implies a net influx

of resources for the central bank. At the same time a fraction of net income

is devoted to pay dividends on this capital.21 Moreover the central bank

also has operating expenses. We ignore these issues in computing the sim-

ulated path of remittances since it would further complicate the description

of the remittance rule, and also quantitatively they are not very important

in terms of the simulated path for remittances.

In order to compute the path for remittances implied by expression (44)

we need to compute paths forBC

Pand

VP

. For the former, we make assump-

tions about the path of the central bank’s assets. We make two assumptions

about the future path of BC:

BC

P=

−(δ + n)

BC

P(1)

−(δ + n + s)BC

P(2)

, for t ≤ T, (45)

where T is the time when the size of reserves has reached the early 2008 level

(adjusted for inflation, population and productivity growth), after whichBC

Pgrows with productivity (i.e.,

BC

Pe−γt is constant over time, yielding

BC

P= (γ +

PP)

BC

P). Under assumption (1) the central bank lets its holdings

of government debt mature, while under assumption (2) it sells its assets at

a rate s per year (we set s = .2). Neither assumption is realistic in the case

of the U.S. (BC has increased in 2013) but the point is to show that different

21In the U.S. the central bank’s capital is a fixed fraction of the capital of the mem-

ber banks, and dividends are 6% of capital (see Carpenter, Ihrig, Klee, Quinn, and Boote

(2013)). Note also that according to our notation dividends are included in τC, this quantity

being the total amount of resources leaving the central bank in any given period. In this

sense referring to τC as “remittances” to the fiscal authority is not entirely appropriate.


future paths for sales can yield quite different paths for τC in the short run,

even though the present value of resources remitted to the fiscal authority

τC is the same. Given remittances and the path forBC

Pwe use the budget

constraint (24) to compute the evolution of reserves in real terms:

.(VP

)= (r− n− P

P)

VP

− (χ + δ− (n + δ)q)BC

P+ q

BC

P−(

n +MM

)MP

+ τC. (46)

A legitimate question (also posed by Hall and Reis (2013)) is whether

a rule like (44) keeps the central bank’s capital measured at market prices,

namely

K =(

qBC −V −M)

ent. (47)

stationary. Using the budget constraint (24) we write the evolution of de-

trended capital in real terms (KP

e−(γ+n)t) as

d

(KP

−(γ+n)t)

= (ρ− n− γ)

(KP

e−(γ+n)t)+

(r

MP− τC

)e−γt (48)

So the rule that stabilizesKP

e−(γ+n)t is

τC = rMP

+ (ρ− n− γ)

(KP

e−(γ+n)t)

= r(

qBP− V

P

)−(

PP+ n + γ

)(KP

e−(γ+n)t)

. (49)

This rule has quite different implications for remittances relative to the rule

(44) outside of steady state, but at steady state the two coincide. In fact, at

steady state q = q = 1, K = K, and χ = r, hence the term r(

qBP− V

P

)coincides with the right hand side of expression (44). The remaining term

(π + n + γ)

(KP

e−(γ+n)t)

accounts for the fact that capital increases with

inflation, productivity, and population growth (as discussed above), a factor


which we ignore in (44). If we did properly account for it, expressions (49)

and (44) would be consistent with each other at least at steady state.


TABLE 1. Change in steady state after 2% inflation scare

r v C M/P log(P) log(M) x dpvs

initial 0.015 12.247 1.014 0.083 0.000 0.000 0.000 0.000

new 0.075 27.386 1.012 0.037 0.080 -0.726 0.020 2.667

duration 2.5 5 10 20

A0 = 3 A loss -0.14 -0.28 -0.52 -0.92

gap -0.22 0.35 1.13 1.98

duration 2.5 5 10 20

A0 = 6 A loss -0.14 -0.28 -0.52 -0.92

gap 0.57 1.71 3.25 4.95


TABLE 2. Parameters

normalization, foreign assets

Y− G = 1 F0 = 0

initial assets, reserves, and currencyBC

P= 0.327

VP

= .207MP

= .104

discount rate, reversion to st.st., population and productivity growth

β = 0.01 γ = 0.01

ϕ1 = 0.750 n = 0.0075

monetary policy

θπ = 2 θr = 1

π = 0.02

money demand

ψ0 = 2.31*10−6 ψ1 = -0.055bonds: duration and coupon

δ−1 = 6 χ = 0.035


TABLE 3. Central bank’s resources under different simulations

(1) (2) (3) (4) (5) (6) (7)

qB/P−V/P

PDVseigniorage (1)+(2)

τC

(averageremittance)

q ∆M/P B/B

Baseline calibration

(1) Baseline scenario 0.146 0.998 1.144 0.0026 1.08

(2) Higher rates (β) 0.133 0.169 0.302 0.0033 1.06 -0.006 14.42

(3) Higher rates (γ) 0.142 1.287 1.429 0.0031 1.06 0.003 64.59

(4) Higher rates (β)/same intercept 0.097 0.241 0.338 0.0037 1.01 -0.025 5.75

(5) Higher rates (γ)/same intercept 0.105 1.573 1.677 0.0037 1.01 -0.017 24.56

(6) Inflation scare 0.068 1.007 1.075 0.0024 0.92 -0.024 7.81

(7) Explosive path 3.124 Inf Inf Inf 0.88 3.043 Inf

Higher θπ


(9) Inflation scare 0.085 0.992 1.077 0.0024 0.96 -0.021 10.10


Lower θπ

(11) Higher rates (β)/same intercept -0.055 1.162 1.107 0.0132 0.52 -0.024 3.38

(12) Inflation scare -0.012 1.038 1.026 0.0025 0.67 -0.021 3.90



TABLE 4. Central bank’s resources under different simula-

tions. Calibration with money demand = 0 for r > 100%

(1) (2) (3) (4) (5) (6) (7)

qB/P−V/P

PDVseigniorage (1)+(2)

τC

(averageremittance)

q ∆M/P B/B

Baseline calibration

(1) Baseline scenario 0.146 0.214 0.360 0.0008 1.08


(3) Inflation scare 0.012 0.215 0.227 0.0005 0.92 -0.080 2.44

(4) Explosive path 0.079 -0.094 -0.015 -0.0000 0.88 -0.002 0.92

Higher θπ


(6) Inflation scare 0.029 0.121 0.150 0.0003 0.96 -0.077 2.27

(7) Explosive path 0.001 -0.097 -0.096 -0.0002 0.64 -0.002 0.78

Lower θπ

(8) Higher rates (β)/same intercept -0.111 0.095 -0.016 -0.0002 0.52 -0.080 0.97

(9) Inflation scare -0.072 0.299 0.227 0.0005 0.67 -0.081 1.64

(10) Explosive path 0.133 0.041 0.175 0.0004 1.05 -0.002 6.46


FIGURE 1. A scatter plot of short term interest rates and M/PC

0 0.05 0.1 0.15 0.20.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

M/P

C

r

2013

2008

Modelpre 1980 datapost 1980 data


FIGURE 2. Short term interest rates and inflation: baseline vs

higher rates

Nominal Real

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

1

2

3

4

5

6

r

BaselineHigher ratesHigher rates (same intercept)

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

0.5

1

1.5

2

2.5

3

ρ

BaselineHigher rates

Inflation

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023−0.5

0

0.5

1

1.5

2

2.5

3

π

BaselineHigher ratesHigher rates (same intercept)


FIGURE 3. seigniorage and M/PC

Data Model

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.09

0.095

0.1

M/PCE −− left axis

1980 1985 1990 1995 2000 2005 20100

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

S/Y −− right axis

0.06

0.065

0.07

0.075

0.08

0.085

0.09

M/PC −− left axis

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023−8

−6

−4

−2

0

2

4x 10

−3

Seigniorage −− right axis

Notes: Paths for seigniorage and real money balances in right hand panel are obtained underbaseline simulation.

FIGURE 4. Paths for remittances

Baseline Inflation Scare

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20232

3

4

5

6

7

8

9

10x 10

−3

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Notes: Each panel shows remittances under two assumptions for the path of assets BC: under thefirst assumption (solid line) the central bank lets its assets depreciate, while in the second one itactively sells assets at a rate of 20 percent per year. The paths in the left and right panels areobtained under the baseline and “inflation scare” scenario, respectively.


FIGURE 5. The effect of different inflation responses in inter-

est rate rule under different scenarios

Higher β/same interceptShort term interest rates

Inflation scareShort term interest rates

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

5

10

15

20

25

r

Baselineθπ=2

θπ=3

θπ=1.05

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

2

4

6

8

10

12

14

16

18

r

Baselineθπ=2

θπ=3

θπ=1.05

Inflation scareInflation

Explosive pathsShort term interest rates

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023−2

0

2

4

6

8

10

12

14

π

Baselineθπ=2

θπ=3

θπ=1.05

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 20230

5

10

15

20

25

r

Baselineθπ=2

θπ=3

θπ=1.05


FIGURE 6. Case where money demand = 0 for r > 100%

Short term interest rates and M/PC Seigniorage and real money balances

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

M/P

C

r

2012

2008

Modelpre 1980 datapost 1980 data

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

M/PC −− left axis

2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023−0.025

−0.02

−0.015

−0.01

−0.005

0

0.005

Seigniorage −− right axis

Notes: Paths for seigniorage and real money balances in right hand panel are obtained underbaseline simulation.

FIGURE 7. Laffer curves

Baseline Calibration withmoney demand = 0 for r > 100%

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.02

0.04

0.06

0.08

0.1

0.12

Se

ign

iora

ge

piss

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

0.2

0.4

0.6

0.8

1

1.2x 10

−3

Sei

gnio

rage

piss


FIGURE 8. Self-fulfilling solvency crises

Baseline Calibration withmoney demand = 0 for r > 100%

Threshold Balance Sheet Limit% of current B

qB−V

seigniorage

Notes: The figure shows 1) Top panel: the level of the balance sheet (relative to the current level) forwhich multiple equilibria are possible; 2) Middle panel: the level of qB−V as a fraction of incomefor the current balance sheet size under alternative scenarios; 3) Bottom panel: the level ofseigniorage as a fraction of income under alternative scenarios; as a function of a) inflation in therealternative regime (π), b) the duration of the alternative regime (∆). In all simulation the alternativeregime is expected to start after 3 years. The left and right figures are for the baseline and alternativetransaction technology, respectively. .


E-mail address: [email protected]

E-mail address: [email protected]

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